The extremities of the diameter of a circle are (1, 2) and (3, 4). Then its centre is .........., radius .......... and equation is .......... .

Answer

To solve the problem, we need to find the center, radius, and equation of the circle given the extremities of its diameter at points (1, 2) and (3, 4).

### Step-by-Step Solution:

**Step 1: Find the Center of the Circle**
The center of the circle is the midpoint of the diameter. The midpoint (M) can be calculated using the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \((x_1, y_1) = (1, 2)\) and \((x_2, y_2) = (3, 4)\).

Calculating the coordinates:
\[ M = \left( \frac{1 + 3}{2}, \frac{2 + 4}{2} \right) = \left( \frac{4}{2}, \frac{6}{2} \right) = (2, 3) \]
Thus, the center of the circle is at **(2, 3)**.

**Step 2: Find the Radius of the Circle**
The radius is half the distance of the diameter. First, we calculate the distance (d) between the two endpoints of the diameter using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the values:
\[ d = \sqrt{(3 - 1)^2 + (4 - 2)^2} = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
Now, the radius (r) is half of this distance:
\[ r = \frac{d}{2} = \frac{2\sqrt{2}}{2} = \sqrt{2} \]
Thus, the radius of the circle is **√2**.

**Step 3: Write the Equation of the Circle**
The standard form of the equation of a circle is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center and \(r\) is the radius. Substituting the values we found:
\[ h = 2, \quad k = 3, \quad r = \sqrt{2} \]
Thus, the equation becomes:
\[ (x - 2)^2 + (y - 3)^2 = (\sqrt{2})^2 \]
This simplifies to:
\[ (x - 2)^2 + (y - 3)^2 = 2 \]
So, the equation of the circle is **(x - 2)² + (y - 3)² = 2**.

### Summary of Results:
- Center: (2, 3)
- Radius: √2
- Equation: (x - 2)² + (y - 3)² = 2